Properties

Label 16T13
Degree $16$
Order $16$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_{8}$

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Show commands: Magma

magma: G := TransitiveGroup(16, 13);
 

Group invariants

Abstract group:  $D_{8}$
magma: IdentifyGroup(G);
 
Order:  $16=2^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $13$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $16$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,11)(2,12)(3,10)(4,9)(5,8)(6,7)(13,15)(14,16)$, $(1,4,6,8,10,12,14,15)(2,3,5,7,9,11,13,16)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $D_{8}$ x 2

Low degree siblings

8T6 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)$
2B $2^{8}$ $4$ $2$ $8$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,16)$
2C $2^{8}$ $4$ $2$ $8$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,15)(14,16)$
4A $4^{4}$ $2$ $4$ $12$ $( 1, 6,10,14)( 2, 5, 9,13)( 3, 7,11,16)( 4, 8,12,15)$
8A1 $8^{2}$ $2$ $8$ $14$ $( 1, 4, 6, 8,10,12,14,15)( 2, 3, 5, 7, 9,11,13,16)$
8A3 $8^{2}$ $2$ $8$ $14$ $( 1,12, 6,15,10, 4,14, 8)( 2,11, 5,16, 9, 3,13, 7)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A 8A1 8A3
Size 1 1 4 4 2 2 2
2 P 1A 1A 1A 1A 2A 4A 4A
Type
16.7.1a R 1 1 1 1 1 1 1
16.7.1b R 1 1 1 1 1 1 1
16.7.1c R 1 1 1 1 1 1 1
16.7.1d R 1 1 1 1 1 1 1
16.7.2a R 2 2 0 0 2 0 0
16.7.2b1 R 2 2 0 0 0 ζ81ζ8 ζ81+ζ8
16.7.2b2 R 2 2 0 0 0 ζ81+ζ8 ζ81ζ8

magma: CharacterTable(G);
 

Regular extensions

Data not computed